Subtract the following rational expressions. $\dfrac{9m-1}{8m+2}-\dfrac{5}{7m^2}=$
We can subtract two rational expressions whose denominators are equal by subtracting the numerators and keeping the denominator the same. [Does this fit with how we subtract rational numbers?] When the denominators are not the same, we must manipulate them so that they become the same. In other words, we must find a common denominator. Since the two denominators do not share any common factors, the common denominator is simply the product of these two denominators: $({8m+2})\cdot({7m^2})$. Let's manipulate the expressions to have that denominator: $\begin{aligned} &\phantom{=}\dfrac{9m-1}{{8m+2}}-\dfrac{5}{{7m^2}} \\\\ &=\dfrac{(9m-1)\cdot({7m^2})}{({8m+2})\cdot({7m^2})}-\dfrac{5\cdot({8m+2})}{({7m^2})\cdot({8m+2})} \end{aligned}$ [Why did we do that?] Now that both denominators are the same, let's subtract! $\begin{aligned} &\phantom{=}\dfrac{(9m-1)\cdot(7m^2)}{(8m+2)\cdot(7m^2)}-\dfrac{5\cdot(8m+2)}{(7m^2)\cdot\!(8m+2)} \\\\ &=\dfrac{(9m-1)\cdot(7m^2)-5\cdot(8m+2)}{(8m+2)(7m^2)} \\\\ &=\dfrac{63m^3-7m^2-40m-10}{(8m+2)(7m^2)} \end{aligned}$ In conclusion, $\dfrac{9m-1}{8m+2}-\dfrac{5}{7m^2}=\dfrac{63m^3-7m^2-40m-10}{(8m+2)(7m^2)}$